Group #26143440 — subgroups

Subgroups — #26143440

Order 12032.

3998 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 3998 conjugacy classes of subgroups — too many to draw. Only its 651 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688986	C₂	1	6	yes	maximal
2	#20241659	C₂	1	8	yes	maximal, complemented
3	C₂⁴×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
5	#15688986	—	2	24	no	—
6	C₂×C₄₇⋊#95	C₂²	1	24	yes	—
7	C₂³×Q₈×C₄₇	—	2	24	no	—
8	C₂³×Q₈×C₄₇	C₄	1	8	yes	complemented
9	C₂³×Q₈×C₄₇	C₂²	1	4	yes	—
10	C₂⁴×C₄×C₄₇	C₂²	1	3	yes	—
11	C₂³×Dic₄₇	C₂³	1	8	yes	—
12	C₂×C₂²⋊Dic₄₇	—	2	24	no	—
13	C₄₇⋊#95	—	2	48	no	—
14	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
15	C₂²×Q₈×C₄₇	—	2	128	no	—
16	C₂²×Q₈×C₄₇	—	2	96	no	—
17	C₂²×Q₈×C₄₇	—	2	48	no	—
18	C₂²×Q₈×C₄₇	C₂×C₄	1	16	yes	—
19	C₂³×C₄×C₄₇	—	2	24	no	—
20	C₂³×C₄×C₄₇	C₂×C₄	1	12	yes	—
21	C₂³×C₄×C₄₇	D₄	1	24	yes	—
22	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
23	C₂⁴×C₄₇	—	2	8	no	—
24	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
25	C₂⁴×C₄₇	C₂×D₄	1	12	yes	—
26	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
27	C₂×Q₈×C₄₇	—	2	128	no	—
28	C₂×Q₈×C₄₇	—	2	256	no	—
29	C₂×Q₈×C₄₇	—	2	96	no	—
30	C₂²×C₄×C₄₇	—	2	48	no	—
31	C₂²×C₄×C₄₇	—	2	96	no	—
32	C₂²×C₄×C₄₇	—	2	24	no	—
33	C₂²×C₄×C₄₇	C₂²⋊C₄	1	48	yes	—
34	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	24	yes	—
36	C₂²×Dic₄₇	—	2	24	no	—
37	C₂²×Dic₄₇	—	2	32	no	—
38	C₂²⋊Dic₄₇	—	4	16	no	—
39	C₂³×C₄₇	—	2	8	no	—
40	C₂³×C₄₇	—	2	48	no	—
41	C₂³×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
42	C₂³×C₄₇	#120	1	16	yes	—
43	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
44	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
45	C₂³×C₄₇	2₋¹⁺⁴	1	8	yes	—
46	Q₈×C₄₇	—	2	128	no	—
47	C₂×C₄×C₄₇	—	2	24	no	—
48	C₂×C₄×C₄₇	—	2	48	no	—
49	C₂×C₄×C₄₇	—	2	96	no	—
50	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	24	yes	—
51	C₂×Dic₄₇	—	4	8	no	—
52	C₂×Dic₄₇	—	4	48	no	—
53	#14336	—	47	1	no	maximal, Sylow
54	C₂²×C₄₇	—	2	24	no	—
55	C₂²×C₄₇	—	2	32	no	—
56	C₂²×C₄₇	#379	1	8	yes	—
57	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	3	yes	—
58	C₂²×C₄₇	#519	1	4	yes	—
59	C₂²×C₄₇	C₂×#120	1	4	yes	—
60	C₂²×C₄₇	#536	1	24	yes	—
61	C₄×C₄₇	—	2	24	no	—
62	Dic₄₇	—	8	8	no	—
63	C₂⁴×Q₈	D₄₇	1	1	yes	characteristic
64	C₂×#381	—	47	6	no	—
65	#1446	—	47	8	no	—
66	C₂×C₄₇	—	2	8	no	—
67	C₂×C₄₇	C₂×#379	1	1	yes	characteristic
68	C₂×C₄₇	#2310	1	6	yes	—
69	C₂×C₄₇	#2614	1	8	yes	—
70	C₂³×Q₈	—	2	24	no	—
71	C₂³×Q₈	Dic₄₇	1	8	yes	complemented
72	C₂³×Q₈	C₂×D₄₇	1	4	yes	—
73	C₂⁴×C₄	C₂×D₄₇	1	3	yes	—
74	C₂²×C₂²⋊C₄	—	47	4	no	—
75	#381	—	94	24	no	—
76	C₂×#95	—	47	24	no	—
77	C₄₇	#14336	1	1	yes	characteristic, complemented, Sylow
78	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
79	C₂²×Q₈	—	2	128	no	—
80	C₂²×Q₈	—	2	96	no	—
81	C₂²×Q₈	—	2	48	no	—
82	C₂²×Q₈	C₂×Dic₄₇	1	16	yes	—
83	C₂³×C₄	—	2	24	no	—
84	C₂³×C₄	—	47	8	no	—
85	C₂³×C₄	C₂×Dic₄₇	1	12	yes	—
86	C₂³×C₄	C₂²⋊D₄₇	1	24	yes	—
87	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
88	C₂×C₂²⋊C₄	—	94	24	no	—
89	#95	—	94	48	no	—
90	C₂⁴	—	2	8	no	—
91	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
92	C₂⁴	C₂×C₂²⋊D₄₇	1	12	yes	—
93	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
94	C₂×Q₈	—	2	256	no	—
95	C₂×Q₈	—	2	128	no	—
96	C₂×Q₈	—	2	96	no	—
97	C₂²×C₄	—	2	48	no	—
98	C₂²×C₄	—	2	96	no	—
99	C₂²×C₄	—	2	24	no	—
100	C₂²×C₄	—	94	24	no	—
101	C₂²×C₄	—	94	32	no	—
102	C₂²×C₄	C₂²⋊Dic₄₇	1	48	yes	—
103	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
104	C₂²×C₄	C₂×C₂²⋊D₄₇	1	24	yes	—
105	C₂²⋊C₄	—	188	16	no	—
106	C₂³	—	2	8	no	—
107	C₂³	—	2	48	no	—
108	C₂³	C₂×C₂²⋊Dic₄₇	1	12	yes	—
109	C₂³	#13093993	1	16	yes	—
110	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
111	C₂³	C₂²×C₂²⋊D₄₇	1	6	yes	—
112	C₂³	#13094072	1	8	yes	—
113	Q₈	—	2	128	no	—
114	C₂×C₄	—	2	96	no	—
115	C₂×C₄	—	2	24	no	—
116	C₂×C₄	—	2	48	no	—
117	C₂×C₄	—	188	8	no	—
118	C₂×C₄	—	188	48	no	—
119	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	24	yes	—
120	C₂²	—	2	32	no	—
121	C₂²	—	2	24	no	—
122	C₂²	#15688979	1	8	yes	—
123	C₂²	C₂²×C₂²⋊Dic₄₇	1	3	yes	—
124	C₂²	#15689601	1	4	yes	—
125	C₂²	C₂×#13093993	1	4	yes	—
126	C₂²	#15689697	1	24	yes	—
127	C₄	—	2	24	no	—
128	C₄	—	376	8	no	—
129	C₂	—	2	8	no	—
130	C₂	C₂×#15688979	1	1	yes	characteristic
131	C₂	#20245974	1	6	yes	—
132	C₂	#20249107	1	8	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688986

C₂

yes

maximal

#20241659

C₂

yes

maximal, complemented

C₂⁴×Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₂²⋊Dic₄₇

C₂²

yes

—

#15688986

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂³×Q₈×C₄₇

—

C₂³×Q₈×C₄₇

C₄

yes

complemented

C₂³×Q₈×C₄₇

C₂²

yes

—

C₂⁴×C₄×C₄₇

C₂²

yes

—

C₂³×Dic₄₇

C₂³

yes

—

C₂×C₂²⋊Dic₄₇

—

C₄₇⋊#95

—

C₂⁵×C₄₇

C₂³

yes

characteristic

C₂²×Q₈×C₄₇

—

128

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

—

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

D₄

yes

—

C₂³×C₄×C₄₇

C₂³

yes

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

C₂²×C₄

yes

—

C₂⁴×C₄₇

C₂×D₄

yes

—

C₂⁴×C₄₇

C₂⁴

yes

characteristic

C₂×Q₈×C₄₇

—

128

—

C₂×Q₈×C₄₇

—

256

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂³×C₄₇

#120

yes

—

C₂³×C₄₇

C₂³×C₄

yes

characteristic

C₂³×C₄₇

C₂²×D₄

yes

—

C₂³×C₄₇

2₋¹⁺⁴

yes

—

Q₈×C₄₇

—

128

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

#14336

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#379

yes

—

C₂²×C₄₇

C₂²×C₂²⋊C₄

yes

—

C₂²×C₄₇

#519

yes

—

C₂²×C₄₇

C₂×#120

yes

—

C₂²×C₄₇

#536

yes

—

C₄×C₄₇

—

Dic₄₇

—

C₂⁴×Q₈

D₄₇

yes

characteristic

C₂×#381

—

#1446

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#379

yes

characteristic

C₂×C₄₇

#2310

yes

—

C₂×C₄₇

#2614

yes

—

C₂³×Q₈

—

C₂³×Q₈

Dic₄₇

yes

complemented

C₂³×Q₈

C₂×D₄₇

yes

—

C₂⁴×C₄

C₂×D₄₇

yes

—

C₂²×C₂²⋊C₄

—

#381

—

C₂×#95

—

C₄₇

#14336

yes

characteristic, complemented, Sylow

C₂⁵

C₂²×D₄₇

yes

characteristic

C₂²×Q₈

—

128

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×Dic₄₇

yes

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×Dic₄₇

yes

—

C₂³×C₄

C₂²⋊D₄₇

yes

—

C₂³×C₄

C₂²×D₄₇

yes

—

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

—

C₂⁴

C₂²×Dic₄₇

yes

—

C₂⁴

C₂×C₂²⋊D₄₇

yes

—

C₂⁴

C₂³×D₄₇

yes

characteristic

C₂×Q₈

—

256

—

C₂×Q₈

—

128

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

103

C₂²×C₄

C₂²×Dic₄₇

yes

—

104

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

105

C₂²⋊C₄

—

188

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂×C₂²⋊Dic₄₇

yes

—

109

C₂³

#13093993

yes

—

110

C₂³

C₂³×Dic₄₇

yes

characteristic

111

C₂³

C₂²×C₂²⋊D₄₇

yes

—

112

C₂³

#13094072

yes

—

113

Q₈

—

128

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

188

—

118

C₂×C₄

—

188

—

119

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

#15688979

yes

—

123

C₂²

C₂²×C₂²⋊Dic₄₇

yes

—

124

C₂²

#15689601

yes

—

125

C₂²

C₂×#13093993

yes

—

126

C₂²

#15689697

yes

—

127

C₄

—

128

C₄

—

376

—

129

C₂

—

130

C₂

C₂×#15688979

yes

characteristic

131

C₂

#20245974

yes

—

132

C₂

#20249107

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂⁴×Q₈×C₄₇	C₂
socle, cosocle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂⁵×C₄₇	C₂³
hypercenter	C₂⁴×Q₈	D₄₇
hypocenter	C₄₇	#14336
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×Q₈

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₄₇

commutator

C₂³×C₄₇

C₂³×C₄

Fitting

C₂⁴×Q₈×C₄₇

C₂

socle, cosocle

C₂⁴×C₄₇

C₂⁴

prime radical

C₂⁵×C₄₇

C₂³

hypercenter

C₂⁴×Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂⁴ ◁ C₂⁴×Q₈

N	H ≅ G/N	type
2. #20241659	C₂	semidirect
8. C₂³×Q₈×C₄₇	C₄	semidirect
71. C₂³×Q₈	Dic₄₇	semidirect
77. C₄₇	#14336	semidirect

H ≅ G/N

type

2. #20241659

C₂

semidirect

8. C₂³×Q₈×C₄₇

C₄

semidirect

71. C₂³×Q₈

Dic₄₇

semidirect

77. C₄₇

#14336

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15688986	2
#20241659	2
C₂⁴×Q₈×C₄₇	2
#14336	47

subgroup

index

C₂×#15688986

#20241659

C₂⁴×Q₈×C₄₇

#14336

Subgroups — #26143440

Conjugacy classes of subgroups

Complements

Maximal subgroups